Why is this important ?1. Because the continued fraction algorithm embodies hierarchical , hyperbolic power principles . A term far to the right of the expression has little effect on the value of x . But the effect is also of the form y=1/z (hyperbolic , not linear) . We have thus a computational method embodying non-linear and hierarchical principles .

2. Khinchin's Constant . K= 2.68545…See www.mathworld.com " Continued Fractions" for the definition . It means that there is a mathematical Invariant lurking around . Invariant Quantal Tensors can be defined from this (see para 3 below) . Co-variant quantal tensors from Khinchin-Levy Constant ?

3. The Quantal bit .The Cantor diagonal proof is just as applicable here . Orders of Randomness can thus be adduced .

But because the Continued Fraction algorithm incorporates division , we can generate an infinity (actually at least aleph(1) ) number of negative numbers using the diagonal method , each one causing a discontinuity . (ie 1+ a(n) = 0 ) These discontinuities define the quanta .

This makes most of formal differentiation or integration unworkable without some major re-timbering .

But actually , regularized discontinuities can be incorporated into a formal framework without busting the bank .

Sub-Planck Catalysts become a real possibility , even if only using pico- or nano metamaterials .

But numerical methods can (and has been ) used to great success . But pesky summation-to-infinity near discontinuity boundaries will persist .

Beth(x) systems can also be defined from this basis . (See http://andreswhy.blogspot.com "New Tools : Orders of Randomness" )

Particles , humans and the state .These are all identifiable (ie delineated) states . Their boundaries can be described as discontinuities . Therefore we can use the above methods to get a better description than possible with continuous mathematics .

A good indication that we are on the right track is the use of so-called Recurrence Relations in the Wolfram article . These are simply our old friend Fibonacci (ie an iterative growth statement of a finite process . )The form is g(n+1) = g1(n) +g2(n-1) . A term is dependant on the previous two terms . )

Note that the Fibonacci Golden Ratio squared is approximately equal to Khinchin's constant ( 1.618^2 ~ 2.685 ) . Some relationship would be expected .

If we can develop an Invariant Quantal Tensor , we will be able to assign (ie calculate) a number giving a one-dimensional value to the whole spectrum of physical reality , from sub-atomic particles to humans to companies to national states , rating each .

I cannot see this being very popular .

"All persons with a number less than 100 , please report to the Soylent Green Department."